Identification of Parameterized Gray-Box State-Space Systems: From a Black-Box Linear Time-Invariant Representation to a Structured One

Abstract

International audienceWhile determining the order as well as the matrices of a black-box linear state-space model is now an easy problem to solve, it is well-known that the estimated (fully parameterized) state-space matrices are unique modulo a non-singular similarity transformation matrix. This could have serious consequences if the system being identified is a real physical system. Indeed, if the true model contains physical parameters, then the identified system could no longer have the physical parameters in a form that can be extracted easily. By assuming that the system has been identified consistently in a fully parameterized form, the question addressed in this paper then is how to recover the physical parameters from this initially estimated black-box form. Two solutions to solve such a parameterization problem are more precisely introduced. First, a solution based on a null-space-based reformulation of a set of equations arising from the aforementioned similarity transformation problem is considered. Second, an algorithm dedicated to nonsmooth optimization is presented to transform the initial fully parameterized model into the structured state-space parameterization of the system to be identified. A specific constraint on the similarity transformation between both system representations is added to avoid singularity. By assuming that the physical state-space form is identifiable and the initial fully parameterized model is consistent, it is proved that the global solutions of these two optimization problems are unique. The proposed algorithms are presented, along with an example of a physical system

Identifying Position-Dependent Mechanical Systems: A Modal Approach Applied to a Flexible Wafer Stage

“…This transformation performs well for the considered system, as shown by the results in Section IV-B. For more general approaches to transform blackbox models to a gray-box models, see, e.g., [59].…”

Subspace identification of 1D large-scale heterogeneous network

“…From the matrix quantities in (18) and (19), we can see that structured matrices O 2R+1,s/2 and C 2R+1,s/2 are linearly parameterized by the block components of W s/2 and E s/2 , respectively. By the result of Theorem 2, it can be derived that the optimal solution to (20) can yield the estimate of W s/2 , E s/2 up to an ambiguity matrix.…”

“…Second is the retrieval of the system matrices, describing the local LTI dynamics, of an individual subsystem and its interaction with its neighboring systems (up to a similarity transformation) from the reliably estimated Markov parameters. The retrieval of the system matrices of an individual subsystem is inherently a challenging structured state-space realization problem [18] for which the optimal solution can yield the estimates of system matrices up to a similarity transformation. The second feature also includes the exploitation of the shifting structure of a timevarying generalized observability matrix.…”

System Identification Algorithm for Computing the Modal Parameters of Linear Mechanical Systems

“…The methods of system identification can be applied to deterministic systems, as well as to stochastic processes [28,29]. Furthermore, several effective system identification methods have been developed in the literature such as, for example, adaptive algorithms, least square methods, gradient-based procedures and multi-innovation identification methodologies [30][31][32][33][34][35]. In engineering applications, the methods of system identification are principally used for performing the experimental modal analysis, which represents the principal topic of interest of this investigation.…”